What is power anyway?

Some of the wording around power can be quite confusing. For example, the power of a two-sample t-test is:

The probability of rejecting the null in favor of the alternative, assuming there is indeed a difference in population means between the two groups.

That's quite a mouthful!

Imagine you are analyzing a marketing campaign and want to infer if it increases customer spending. You conduct a two-sample t-test (customers exposed to the campaign versus customers not exposed) to compare the mean dollar amounts spent for each group.